Dihedral and cyclic symmetric maps on surfaces

TL;DR

This paper classifies semi-equivelar maps on higher genus surfaces, showing they have symmetry groups isomorphic to dihedral or cyclic groups, and identifies 39 types on a specific surface with Euler characteristic -2.

Contribution

It proves the existence of at least 39 types of semi-equivelar maps with specific symmetry groups on certain surfaces and extends the classification to include one additional map type.

Findings

At least 39 types of semi-equivelar maps on surfaces with Euler characteristic -2m.

Symmetry groups of these maps are isomorphic to dihedral or cyclic groups.

Complete classification for some types, with extensions to additional types.

Abstract

If the face\mbox{-}cycles at all the vertices in a map are of the same type, then the map is said to be a semi-equivelar map. Automorphism (symmetry) of a map can be thought of as a permutation of the vertices which preserves the vertex\mbox{-}edge\mbox{-}face incidences in the embedding. The set of all symmetries forms the symmetry group. In this article, we discuss the maps' symmetric groups on higher genus surfaces. In particular, we show that there are at least $39$ types of the semi-equivelar maps on the surface with Euler char. $-2m, m \ge 2$ and the symmetry groups of the maps are isomorphic to the dihedral group or cyclic group. Further, we prove that these $39$ types of semi-equivelar maps are the only types on the surface with Euler char. $-2$. Moreover, we know the complete list of semi-equivelar maps (up to isomorphism) for a few types. We extend this list to one more type and can classify others similarly. We skip this part in this article.